In Trinitarian apologetics, as part of efforts to establish the doctrine of the Trinity in the Old Testament, appearances of the Angel of the Lord are commonly appealed to as instances of the pre-incarnate Jesus appearing, and being identified as Yahweh, or as otherwise demonstrating in some way that there is plurality within Yahweh. Commonly appealed to passages include Genesis 16:7-13, 18:1-33, 22:11-18, 32:24-30, 48:15-16, Exodus 3:1-10, 14:19, 23:20-23, Numbers 22:31-38, Joshua 5:13-15, Judges 2:1-3, 6:11-24, 13:3-22, in which the Angel of the Lord is supposed to be mentioned, and identified with Yahweh in some form.
Although the circumstances will differ in each account, a response from a Unitarian point of view to the identity of Angel of the Lord would be:
Therefore, in either case, there would be no need to posit or suggest "another" Yahweh, who is in some way distinct from the Yahweh spoken of in the rest of Scripture, such that the Bible is actually hinting that there is a plurality within God. There is only one God, whose name is Yahweh, and when that is kept in mind, there are really only a few options for how these passages can be reasonably understood. Suggesting extra or distinct Yahwehs is not an interpretive option, but rather, is just reading confusion into these authors.
Additionally, it should be noted that no New Testament author ever appeals to these instances as manifestations of Jesus, nor does Jesus Himself ever say that He was there, or that it was Him, even when speaking on one of the passages in question (Mark 12:26, Luke 20:37-38). If God had intended to reveal that He is multi-personal through these appearances, it would be strange that neither Jesus nor His Apostles ever pointed to them to make that case.
In summary, there are simply too many better, more reasonable ways to understand the Angel of the Lord than to say that it is the Hebrew Bible's way of teaching that God is Triune, or that there exists a plurality within God.